Toward a structure theory of crossing-critical graphs Zdenek Dvo r - PowerPoint PPT Presentation

Toward a structure theory of crossing-critical graphs Zdenek Dvoˇ r´ ak, Petr Hlinˇ en´ y, and Bojan Mohar Simon Fraser University & IMFM Ghent Graph Theory Workshop on Structure and Algorithms 12–14 August 2019 Dvoˇ r´ ak-Hlinˇ en´ y-Mohar Crossing-Critical Structure

Crossing number of a graph Crossing number of G . . . cr ( G ) Minimum number of crossings of edges when G is drawn in the plane cr ( K 5 ) = 1, cr ( K 6 ) ≤ 3 Dvoˇ r´ ak-Hlinˇ en´ y-Mohar Crossing-Critical Structure

Do we understand it well? Is the crossing number equal to the pair-crossing number? Dvoˇ r´ ak-Hlinˇ en´ y-Mohar Crossing-Critical Structure

Do we understand it well? Despite many breakthrough results about crossing numbers, some very basic questions are still unresolved. Dvoˇ r´ ak-Hlinˇ en´ y-Mohar Crossing-Critical Structure

Do we understand it well? Despite many breakthrough results about crossing numbers, some very basic questions are still unresolved. Conjecture (Hill, cca. 1958) cr ( K n ) = H ( n ) := 1 4 ⌊ n 2 ⌋ ⌊ n − 1 2 ⌋ ⌊ n − 2 2 ⌋ ⌊ n − 3 2 ⌋ Dvoˇ r´ ak-Hlinˇ en´ y-Mohar Crossing-Critical Structure

Do we understand it well? Despite many breakthrough results about crossing numbers, some very basic questions are still unresolved. Conjecture (Hill, cca. 1958) cr ( K n ) = H ( n ) := 1 4 ⌊ n 2 ⌋ ⌊ n − 1 2 ⌋ ⌊ n − 2 2 ⌋ ⌊ n − 3 2 ⌋ Known to be true for n ≤ 12 (Pan and Richter). Open for K 13 . Dvoˇ r´ ak-Hlinˇ en´ y-Mohar Crossing-Critical Structure

Do we understand it well? Despite many breakthrough results about crossing numbers, some very basic questions are still unresolved. Conjecture (Hill, cca. 1958) cr ( K n ) = H ( n ) := 1 4 ⌊ n 2 ⌋ ⌊ n − 1 2 ⌋ ⌊ n − 2 2 ⌋ ⌊ n − 3 2 ⌋ Known to be true for n ≤ 12 (Pan and Richter). Open for K 13 . Conjecture (Tur´ an, 1944; Zarankiewicz 1960’s) cr ( K n , m ) = Z ( n , m ) := ⌊ n 2 ⌋ ⌊ n − 1 2 ⌋ ⌊ m 2 ⌋ ⌊ m − 1 2 ⌋ Dvoˇ r´ ak-Hlinˇ en´ y-Mohar Crossing-Critical Structure

Do we understand it well? Despite many breakthrough results about crossing numbers, some very basic questions are still unresolved. Conjecture (Hill, cca. 1958) cr ( K n ) = H ( n ) := 1 4 ⌊ n 2 ⌋ ⌊ n − 1 2 ⌋ ⌊ n − 2 2 ⌋ ⌊ n − 3 2 ⌋ Known to be true for n ≤ 12 (Pan and Richter). Open for K 13 . Conjecture (Tur´ an, 1944; Zarankiewicz 1960’s) cr ( K n , m ) = Z ( n , m ) := ⌊ n 2 ⌋ ⌊ n − 1 2 ⌋ ⌊ m 2 ⌋ ⌊ m − 1 2 ⌋ Known to be true for n ≤ 6 and for K 7 , m (7 ≤ m ≤ 10), Woodall (1993), computer-assisted proof. Open for K 9 , 9 and K 7 , 11 ? Dvoˇ r´ ak-Hlinˇ en´ y-Mohar Crossing-Critical Structure

Crossing-Critical Graphs A graph G is c -crossing-critical ( c -CC) if ◮ cr ( G ) ≥ c , and ◮ for every edge e , cr ( G − e ) < c . Dvoˇ r´ ak-Hlinˇ en´ y-Mohar Crossing-Critical Structure

Crossing-Critical Graphs A graph G is c -crossing-critical ( c -CC) if ◮ cr ( G ) ≥ c , and ◮ for every edge e , cr ( G − e ) < c . Some observations about the definition: ◮ We assume there are no vertices of degree 2 or less. ◮ 1-CC graphs are precisely K 5 and K 3 , 3 . ◮ For every c ≥ 2, there are infinitely many c -CC graphs. Dvoˇ r´ ak-Hlinˇ en´ y-Mohar Crossing-Critical Structure

Crossing-Critical Graphs A graph G is c -crossing-critical ( c -CC) if ◮ cr ( G ) ≥ c , and ◮ for every edge e , cr ( G − e ) < c . Some observations about the definition: ◮ We assume there are no vertices of degree 2 or less. ◮ 1-CC graphs are precisely K 5 and K 3 , 3 . ◮ For every c ≥ 2, there are infinitely many c -CC graphs. ◮ Richter and Thomassen (1993): cr ( G ) ≤ 5 2 c + 16 Dvoˇ r´ ak-Hlinˇ en´ y-Mohar Crossing-Critical Structure

Crossing-Critical Graphs A graph G is c -crossing-critical ( c -CC) if ◮ cr ( G ) ≥ c , and ◮ for every edge e , cr ( G − e ) < c . Some observations about the definition: ◮ We assume there are no vertices of degree 2 or less. ◮ 1-CC graphs are precisely K 5 and K 3 , 3 . ◮ For every c ≥ 2, there are infinitely many c -CC graphs. ◮ Richter and Thomassen (1993): cr ( G ) ≤ 5 2 c + 16 ◮ Toth improved the bound to 2 c + 16. Dvoˇ r´ ak-Hlinˇ en´ y-Mohar Crossing-Critical Structure

Crossing-Critical Graphs A graph G is c -crossing-critical ( c -CC) if ◮ cr ( G ) ≥ c , and ◮ for every edge e , cr ( G − e ) < c . Some observations about the definition: ◮ We assume there are no vertices of degree 2 or less. ◮ 1-CC graphs are precisely K 5 and K 3 , 3 . ◮ For every c ≥ 2, there are infinitely many c -CC graphs. ◮ Richter and Thomassen (1993): cr ( G ) ≤ 5 2 c + 16 ◮ Toth improved the bound to 2 c + 16. Conjecture (Richter) If G is c-CC, then cr ( G ) ≤ c + Θ( √ c ) . Dvoˇ r´ ak-Hlinˇ en´ y-Mohar Crossing-Critical Structure

Constructions for a fixed c ◮ ˇ Siran (1984) ◮ Salazar (2003) ◮ Hlineny (2002) Dvoˇ r´ ak-Hlinˇ en´ y-Mohar Crossing-Critical Structure

More general examples Dvoˇ r´ ak-Hlinˇ en´ y-Mohar Crossing-Critical Structure

Toward a global structure Observation: Large c -CC graphs cannot contain a large grid as a minor (and thus have bounded tree-width). Dvoˇ r´ ak-Hlinˇ en´ y-Mohar Crossing-Critical Structure

Toward a global structure Observation: Large c -CC graphs cannot contain a large grid as a minor (and thus have bounded tree-width). Theorem (Hlineny 2003) For every fixed c, the c-CC graphs have bounded path-width, pw ( G ) ≤ 2 2000 c 3 log c . It was conjectured that c -CC graphs also have bounded bandwidth (as suggested by known examples). Dvoˇ r´ ak-Hlinˇ en´ y-Mohar Crossing-Critical Structure

Big Degrees Dvorak & Mohar (2010): There are c -CC graphs with vertices of arbitrarily large degrees. Recent result (SoCG 2019): Large degrees in c -CC graphs are possible for every c ≥ 13 but do not occur when c ≤ 12. Dvoˇ r´ ak-Hlinˇ en´ y-Mohar Crossing-Critical Structure

Tiles Theorem (Bokal, Oporowski, Richter, Salazar 2016) There is a constant k such that every 2-CC graph contains a set of at most k vertices whose removal leaves a graph, each of whose components is a long sequence of 42 types of tiles. Dvoˇ r´ ak-Hlinˇ en´ y-Mohar Crossing-Critical Structure

Local structure – Bands and Fans Consider an optimal drawing of a c -CC graph. Dvoˇ r´ ak-Hlinˇ en´ y-Mohar Crossing-Critical Structure

Local structure – Bands and Fans Consider an optimal drawing of a c -CC graph. Theorem Any optimal drawing of a large c-CC graph contains a crossing-free subdrawing that is isomorphic to a large band or a large fan. Dvoˇ r´ ak-Hlinˇ en´ y-Mohar Crossing-Critical Structure

Main Theorem – Structure and generation Theorem (Dvorak, Hlineny, M., SoCG 2018) (a) For every c ≥ 2 there is a finite set of c-CC graphs F 1 , . . . , F N ( c ) and every other c-CC graph can be obtained from one of these by replicating tiles of bounded size. (b) Moreover, all graphs obtained through the replication process are c-CC. Dvoˇ r´ ak-Hlinˇ en´ y-Mohar Crossing-Critical Structure

About the proof: Tiles as a semigroup Two tiles are q -equivalent if they contain precisely the same linkages of type q . Theorem Composition of tiles forms a finite semigroup with respect to the q-equivalence. Dvoˇ r´ ak-Hlinˇ en´ y-Mohar Crossing-Critical Structure

Simon’s factorization forest A a finite semigroup s a long product, want to express it as a long product in which many factors are repeated. Theorem ∀ A : ∀ f : N → N : ∃ k 0 , n 0 such that every product s = t 1 t 2 · · · t N with N ≥ n 0 , the sequence can be partitioned into substrings, s = S 0 S 1 S 2 . . . S m S m +1 such that (a) m ≥ f ( k ) and each S i ( 1 ≤ i ≤ m) has length at most k, and (b) the product of elements in each S i ( 1 ≤ i ≤ m) is the same idempotent element of A. Dvoˇ r´ ak-Hlinˇ en´ y-Mohar Crossing-Critical Structure

About the proof (simplified explanation) ◮ Suppose G is very large, consider one of its optimal drawings. ◮ Find a long band or fan. ◮ Show there is a lot of repetition of certain short parts. ◮ Find a repeated part that can be reduced (inverse of replication). ◮ Show that the resulting graph is c -CC if and only if G is c -CC. (For this proof to work we need a slightly more complicated replication condition.) Dvoˇ r´ ak-Hlinˇ en´ y-Mohar Crossing-Critical Structure

Questions? Dvoˇ r´ ak-Hlinˇ en´ y-Mohar Crossing-Critical Structure